Add the flour, baking powder, and salt to the bowl of a food processor and pulse to combine. Add the butter and pulse until the mixture forms a coarse meal, 15 to 30 seconds depending on the size of your cubes. Dump the mixture into a medium bowl and add the milk. Mix with a fork until a scrappy dough forms. (If you don't have a food processor, you can cut the butter into the dry ingredients in a mixing bowl with a pastry blender, fork and knife, or your fingers, and then add the milk).

Turn the dough onto the counter and knead three or four times, folding the dough over on itself, until it comes together. If the dough seems too dry, drizzle a tiny bit more milk over the dry areas and knead again once or twice. Press the dough into a disk about 1/2-inch thick. With a biscuit cutter or a jelly jar, cut straight down into 10 or 12 shapes, enough to cover the cobbler (recombine the scraps to make extra biscuits). Arrange the biscuits on top, brush with a little milk, and sprinkle with the remaining sugar.